Spread the dried beans out, remove any stones or damaged beans, then place them in a large bowl or pot and cover with tap water by at least 2 inches. Leave the beans to soak overnight, or for at least 8 hours, until they look plumper and more evenly hydrated.

Pour off the soaking liquid and drain the beans well. The discarded liquid will look cloudy, while the soaked beans should feel firm but swollen.

Heat the oil in a 5.5 to 7 quart soup pot or Dutch oven over medium heat. When the oil shimmers across the bottom of the pot, it is ready for the sausage.

Add the sliced kielbasa and cook for about 5 minutes, turning the pieces as they brown. Stop when the edges turn golden and the sausage releases a sweet, smoky aroma.

Stir in the onion, carrots, celery, and garlic, then cook for about 5 minutes. The vegetables should soften and smell sweet, with browned bits beginning to collect on the bottom of the pot.

Make room in the center of the pot and nestle in the ham hock, keeping it below the rim. The hock should sit securely among the vegetables rather than sticking above the cooking liquid.

Add the drained beans around the ham hock. The mixed legumes should settle into an even layer around the meat.

Pour in 6 cups of water and add the bay leaves, making sure the beans are fully covered. Bring the pot to a boil, then lower the heat to an active simmer and skim away any gray foam that rises at first.

Set the lid partially over the pot and simmer for a total of 2 hours, stirring every 30 to 40 minutes and adding water as needed to keep the beans submerged. Maintain steady, gentle bubbles rather than a rapid boil, and watch for the broth to gradually turn more opaque.

When 30 minutes remain, lower the heat to a gentle simmer, remove the bay leaves, and stir in the tomatoes, mustard, chili powder, and Worcestershire sauce. The broth will turn a deeper red and smell fuller, while the beans continue softening without the acidity slowing them too early.

Turn off the heat and move the ham hock to a bowl until cool enough to handle, then pull away the edible meat and return it to the soup. Discard the bone and any tough gristle so only tender shreds go back into the pot.

Mash a bean against the side of the pot to check for tenderness, then season with salt as needed. A large lima bean should flatten easily, and the broth should look lightly thickened from the softened legumes.